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Theorem bnj1212 29269
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1212.1  |-  B  =  { x  e.  A  |  ph }
bnj1212.2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
Assertion
Ref Expression
bnj1212  |-  ( th 
->  x  e.  A
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ch( x)    th( x)    ta( x)    B( x)

Proof of Theorem bnj1212
StepHypRef Expression
1 bnj1212.1 . . 3  |-  B  =  { x  e.  A  |  ph }
21bnj21 29180 . 2  |-  B  C_  A
3 bnj1212.2 . . 3  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
43simp2bi 974 . 2  |-  ( th 
->  x  e.  B
)
52, 4bnj1213 29268 1  |-  ( th 
->  x  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1727   {crab 2715
This theorem is referenced by:  bnj1204  29479  bnj1296  29488  bnj1415  29505  bnj1421  29509  bnj1442  29516  bnj1452  29519  bnj1489  29523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-in 3313  df-ss 3320
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